๐ขAlgebraic Number Theory Unit 7 โ Prime Decomposition in Number Fields
Number fields extend rational numbers to include roots of polynomials with rational coefficients. They're finite extensions of rational numbers, where every element is a root of a polynomial with rational coefficients. The ring of integers in a number field consists of elements that are roots of monic polynomials with integer coefficients.
Prime decomposition in number fields generalizes prime factorization in integers. In a number field, prime ideals take the role of prime numbers. Every nonzero ideal can be uniquely expressed as a product of prime ideals. The decomposition of a prime number in the ring of integers is determined by the factorization of its ideal.
Study Guides for Unit 7 โ Prime Decomposition in Number Fields
Number fields extend the concept of rational numbers to include roots of polynomials with rational coefficients
A number field K is a finite extension of the field of rational numbers Q
Every element in K can be expressed as a linear combination of a finite basis over Q
The degree of a number field [K:Q] is the dimension of K as a vector space over Q
Number fields are algebraic extensions of Q, meaning every element is a root of a polynomial with rational coefficients
The ring of integers OKโ of a number field K consists of elements that are roots of monic polynomials with integer coefficients
OKโ is a subring of K and plays a crucial role in studying prime decomposition
Prime Decomposition Basics
Prime decomposition in number fields generalizes the concept of prime factorization in the integers
In a number field K, prime ideals take the role of prime numbers in the ring of integers OKโ
Every nonzero ideal in OKโ can be uniquely expressed as a product of prime ideals (up to the order of factors)
The decomposition of a prime number pโZ in OKโ is determined by the factorization of the ideal pOKโ
pOKโ=p1e1โโโฏprerโโ, where piโ are prime ideals and eiโ are positive integers
The exponents eiโ in the prime decomposition are called the ramification indices
The residue class degree fiโ of a prime ideal piโ is the degree of the field extension OKโ/piโ over Z/pZ
Unique Factorization in Number Fields
Unique factorization of elements in a number field K depends on the properties of its ring of integers OKโ
A number field K is called a unique factorization domain (UFD) if every nonzero element in OKโ can be uniquely expressed as a product of irreducible elements (up to the order and units)
Not all number fields are UFDs; the failure of unique factorization is related to the presence of non-principal ideals in OKโ
The class number hKโ of a number field K measures the extent to which unique factorization fails in OKโ
hKโ=1 if and only if OKโ is a UFD
Examples of number fields that are UFDs include the Gaussian integers Z[i] and the Eisenstein integers Z[ฯ], where ฯ is a primitive third root of unity
Ramification and Splitting
Ramification occurs when a prime ideal p in the ring of integers OKโ appears with an exponent e>1 in the prime decomposition of pOKโ
The prime p is said to ramify in K, and p is called a ramified prime
Splitting refers to the factorization of a prime ideal pOKโ into distinct prime ideals in OKโ
If pOKโ=p1โโฏprโ with distinct prime ideals piโ, then p is said to split completely in K
The splitting behavior of primes in a number field is determined by the Dedekind-Kummer theorem
It relates the splitting of primes to the factorization of certain polynomials modulo p
Ramification is connected to the discriminant ฮKโ of the number field K
A prime p ramifies in K if and only if p divides ฮKโ
The study of ramification and splitting is crucial for understanding the arithmetic properties of number fields and their extensions
Ideal Theory in Number Fields
Ideal theory plays a central role in the study of number fields and their rings of integers
An ideal I in the ring of integers OKโ is a subset closed under addition and multiplication by elements of OKโ
Principal ideals are generated by a single element ฮฑโOKโ, denoted as (ฮฑ)={ฮฑฮฒ:ฮฒโOKโ}
The ideal class group ClKโ of a number field K is the quotient group of fractional ideals modulo principal ideals
Its order is the class number hKโ, which measures the failure of unique factorization in OKโ
The norm of an ideal I is defined as N(I)=โฃOKโ/Iโฃ, the size of the quotient ring
For a principal ideal (ฮฑ), the norm is equal to โฃNK/Qโ(ฮฑ)โฃ, where NK/Qโ is the field norm
Ideal theory allows for a generalization of the unique factorization theorem to all number fields
Every nonzero ideal in OKโ can be uniquely expressed as a product of prime ideals
Dedekind Domains and Prime Ideals
The ring of integers OKโ of a number field K is a Dedekind domain
Dedekind domains are integral domains where every nonzero ideal can be uniquely factored into a product of prime ideals
In a Dedekind domain, prime ideals are maximal ideals, and every nonzero prime ideal is maximal
The localization of a Dedekind domain at a prime ideal is a discrete valuation ring
This property allows for the definition of valuations and completions of number fields
The prime ideals in OKโ lie above prime numbers in Z
The lying above relation is characterized by the prime decomposition of pOKโ for prime numbers p
The Dedekind-Kummer theorem provides a criterion for the splitting behavior of prime ideals in terms of the factorization of polynomials modulo p
The study of prime ideals in Dedekind domains is fundamental to understanding the arithmetic and geometric properties of number fields
Applications and Examples
Fermat's Last Theorem: The proof by Andrew Wiles relies heavily on the theory of elliptic curves and modular forms over number fields
Cryptography: Number fields and their prime ideals are used in various cryptographic schemes, such as the Buchmann-Williams key exchange and the Gentry-Szydlo algorithm
Class field theory: It describes the abelian extensions of a number field in terms of its ideal class group and idele class group
The Hilbert class field of K is the maximal unramified abelian extension of K
Diophantine equations: Many Diophantine equations, such as the Pell equation and the Thue equation, can be studied using the arithmetic of number fields
Algebraic number theory: Prime decomposition is a fundamental tool in the study of zeta functions, L-functions, and arithmetic geometry
Examples of number fields:
Quadratic fields: Q(dโ), where d is a squarefree integer (e.g., Q(2โ), Q(โ5โ))
Cyclotomic fields: Q(ฮถnโ), where ฮถnโ is a primitive n-th root of unity (e.g., Q(i), Q(ฯ))
Advanced Topics and Open Problems
Iwasawa theory: It studies the behavior of prime ideals and class groups in infinite towers of number fields, such as Zpโ-extensions
Langlands program: It seeks to unify various areas of mathematics, including number theory, representation theory, and harmonic analysis, through the study of automorphic forms and Galois representations
Stark conjectures: They relate the values of L-functions to the arithmetic of number fields, providing a generalization of the analytic class number formula
ABC conjecture: It states an inequality involving the prime factors of three relatively prime integers a, b, and c satisfying a+b=c
The conjecture has significant implications for the study of Diophantine equations and the distribution of prime numbers
Fermat-Catalan conjecture: It generalizes Fermat's Last Theorem to other powers and seeks to classify all solutions to the equation am+bn=ck in positive integers with m1โ+n1โ+k1โ<1
Nonabelian class field theory: It aims to extend the results of class field theory to nonabelian extensions of number fields
This area is still largely conjectural and remains an active area of research in modern number theory